multi-jet generation -status report-

Mul$-‐Jet Genera$on -‐ status report -‐

Yoshitaro Takaesu (KEK)

November 16, 2013 1

In collabora$on with Fabio, Johan, Kaoru and Tim

Importance of mul$-‐jet at the LHC

Mul2-‐jet signature appears in many New Physics models.

Event Generator should be able to generate > 4 jets.

November 16, 2013 2

Limita$on of MG Generated Codes (> 8MB) cannot be compiled in usual PC.

gg � 5gqg � q5gqq � qq4g

November 16, 2013 3

8 MB

100

101

102

103

104

105

2 3 4 5 6

log 10

File

Siz

e (K

B)

# of Final State Particles

gg > ngug > ug nguu > uu ng

The file size of codes for QCD processes

How to overcome the Limita$on

November 16, 2013 4

gg � 5gqg � q5gqq � qq4g

Structure of QCD Amp

November 16, 2013 5

gg � gg= �ig2

s [fabef cde(�1 · ��3 �2 · ��4 � �1 · ��4 �2 · ��3)

+ facef bde(�1 · �2 ��3 · ��4 � �1 · ��4 �2 · ��3)

+ fadef bce(�1 · �2 ��3 · ��4 � �1 · ��3 �2 · ��4)]a

b

c

d

fabc = �2iT r�[T a, T b]T c

�Tr(T aT b) =

12�ab

=�

��P (b,c,d)

Tr(T aT �(b)T �(c)T �(d))A(a �(b)�(c)�(d))

= Tr(T aT bT cT d)A(abcd) + Tr(T aT bT dT c)A(abdc)

+ Tr(T aT cT bT d)A(acbd) + Tr(T aT cT dT b)A(acdb)

+ Tr(T aT dT bT c)A(adbc) + Tr(T aT dT cT b)A(adcb)

Color flow

Color Factor color-‐ordered amp.

i: Color flow M =

Ncf�

i=1

CiAi Ncf � (n� 1)!

November 16, 2013 6

�

color

|M|2 =Ncf�

i,j

�

color

C�i CjA

�i Aj

M =Ncf�

i=1

CiAi

STRATEGY

Divide �

color

C�i Cj

A1

A2

A3· · · ANcf�1

ANcf

• Speed up Amp Evalua$on • Off-‐shell Recursive Rela2ons

• Re-‐organize Color Summa$on 1/Nc expansion

Speed up Amp Evalua$on

November 16, 2013 7

November 16, 2013 8

Off-‐shell Recursive Rela$ons

Step #

Time performance

November 16, 2013 9

0

0.05

0.1

0.15

0.2

0.25

0.3

4 5 6 7 8

Tim

e (m

s)

Final Gluons

RecursiveNon-Recursive

n3

n4

November 16, 2013 10

Recursive Rela$ons for gluonic sub-‐amplitude

Re-‐organize Color Sum



1/Nc expansion

Polynomial of Nc:

�

color

|M|2 =Ncf�

i,j

�

color

C�i CjA

�i Aj

= Nmc

�

i

�|Ai|2 +

1N2

cA�

i (Aj1,1 + Aj1,2 + · · · )�

+1

N4c

A�i (Aj2,1 + Aj2,2 + · · · )

+ · · ·

(m = ng + nq/2)

Mul$-‐jet event genera$on

• Generate events with Leading Color Approxima2on

• For each event, include higher order correc2ons into its weight.


�

color

|M|2 =�

i

Nmc

�|Ai|2 +

1N2

cA�

i (Aj1,1 + Aj1,2 + · · · ) +�

1Nc2

�2

A�i (Aj2,1 + Aj2,2 + · · · ) + · · ·

�

�

color

|M|2 =�

i

Nmc

�|Ai|2 +

1N2

cA�

i (Aj1,1 + Aj1,2 + · · · ) +�

1Nc2

�2

A�i (Aj2,1 + Aj2,2 + · · · ) + · · ·

�

� =� �

|M|2d�n

Leading color summa$on

Color-‐flow sampling

Ncf � (m� 1)! ( ~ 40,000 for gg -‐> 7g )


�

color

|M|2 =Ncf�

i=1

Nmc |Ai|2

Too LARGE

Event is a set of momenta, helici$es and a color flow

Results



gg_6g NLO diffEntries 61Mean -0.9229RMS 0.5551

-3 -2.5 -2 -1.5 -1 -0.5 00

2

4

6

8

10

12

14

gg_6g NLO diffEntries 61Mean -0.9229RMS 0.5551

gg_6g NLO diff

= Nmc

�

i

�|Ai|2 +

1N2

cA�

i (Aj1,1 + Aj1,2 + · · · )��

color

|M|2

�

color

|M|2 =Ncf�

i=1

N (n�2)c (N2

c � 1)|Ai|2


ug_u3g NLNc diff

Entries 1000Mean 2.63RMS 2.916

-100 -80 -60 -40 -20 0 20 40 60 80 1000

100

200

300

400

500

ug_u3g NLNc diff

Entries 1000Mean 2.63RMS 2.916

ug_u3g NLNc diff

ug_u3g LNc diffEntries 1000Mean -27.28RMS 13.15

-100 -80 -60 -40 -20 0 20 40 60 80 1000

20

40

60

80

100

120

ug_u3g LNc diffEntries 1000Mean -27.28RMS 13.15

ug_u3g LNc diff

�

color

|M|2 =Ncf�

i=1

(N2c � 1)n�1

Nn�1c

|Ai|2


Event Genera$on @LNc

pp > W + 2 jets W + 3 jets W + 4 jets

MG5 (pb) 10673 4807.9 ± 13.2 2913 ± 7.6

LNc CO (pb) 11076 4623.8 ± 11.9 2460 ± 6

• gg -‐> 3,4,5 g process is now generated without major problem and agree with MG5 (3,4g).

pp > W+2,3,4 jets is also studied recently and roughly agree with MG5.

Preliminary

0.9 1

2 3 4 5 6 7

ratio

Final particles

107

108

109

1010

1011

1012

Cro

ss S

ectio

n (f

b)

gg -> ng Cross Section

CFS (LCA)COMIX

MG5



0

20000

40000

60000

80000

100000

120000

140000

160000

180000

0 50 100 150 200 250 300 350

dX

sec/

dpT1

(pb)

pT1 (GeV)

gg > 4g pT1

MG5 ExactCFsamp LNc

0

5000

10000

15000

20000

25000

30000

35000

-3 -2 -1 0 1 2 3

dX

sec/

dy1

(pb)

y1 (GeV)

gg > 4g y1

MG5 ExactCFsamp LNc

10

100

1000

10000

100000

1e+06

0 50 100 150 200 250 300 350

dX

sec/

dpT1

(pb)

pT1 (GeV)

gg > 4g pT1

MG5 ExactCFsamp LNc

100

1000

10000

100000

0.5 1 1.5 2 2.5 3 3.5 4 4.5 5

dX

sec/

d dR

12 (p

b)

dR12 (GeV)

gg > 4g dR12

MG5 ExactCFsamp LNc

pT1(Linear scale)

pT1(Log scale)

�1 �R12

gg � gggg


dxadxbd�n =2�

s

14�2n�2

�dy1 · · · dyndp2T1 · · · dp2

Tn�1d�1

2�· · · d�n�1

2�

A liee bit of Phase Space Study

Alterna2ve parameteriza2on for MadEvent:

suitable for kinema2cal cuts , e.g., p_T cut or rapidity cut.


2

3

4

5

6

7

8

0 50 100 150 200 250 300 350 400 450 500

MG53chtch

4

5

6

7

-5 -4 -3 -2 -1 0 1 2 3 4 5

MG53chtch

pT1

�1

5

6

7

0 0.5 1 1.5 2 2.5 3 3.5 4 4.5 5

MG53chtch�R23

3

4

5

6

7

8

0 200 400 600 800 1000 1200 1400 1600

MG53chtch

M12

gg � ggg (3.5 % unweigh2ng eff.)

November 16, 2013 23 -4

-3

-2

-1

0

1

0 0.5 1 1.5 2 2.5 3 3.5 4 4.5 5

MG53chtch

-4

-3

-2

-1

0

1

0 200 400 600 800 1000 1200 1400 1600

MG53chtch

-2

-1

0

1

2

800 1000 1200 1400 1600 1800 2000

MG53chtch

-3

-2

-1

0

1

-3 -2 -1 0 1 2 3

MG53chtch

pT1

�1

�R23

M12

gg � ggg

pT1 > 1000GeV


Amplitude

Phase Space

Color Sum

Recursive Rela$ons

Channels

For fast mul$-‐jet genera$on

OutLook • Efficient Phase Space integra2on

• Include higher order correc2ons in 1/Nc.

• Generate events with quarks

• Detailed study for many distribu2ons of Mul2-‐jet events


multi-jet generation -status report-

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